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Logarithmic Corrections to Black HoleEntropy v. 2.0
Finn Larsen
Michigan Center for Theoretical Physics
Recent Developments in String Theory, Ascona, Switzerland, July 24, 2014 .
Logarithmic Corrections
• The leading corrections to the area law for black hole entropy arelogarithmic
δS =1
2D0 logA .
• These corrections can be computed from the low energy theory:only massless fields contribute.
• In some situations the corrections give non-trivial support for aknown microscopic description.
• In other situations they offer clues to the nature of the unknownmicroscopic theory.
2
Updates in v. 2.0In principle: computations are straightforward applications oftechniques from the 70’s.
In recent years, Sen (and collaborators) did what we do, and more.
In practice: computations are cumbersome and intransparent.
Updates in v 2.0 focus on short-cuts that add clarity:
• Interactions with background gravity and graviphoton: employAdS/CFT, specifically organize fluctuations as chiral primaries.
• Contributions from on-shell states only (no ghosts).
• Remnant of unphysical states: simple boundary states .
• Careful with 4D zero-modes (done incorrectly until recent years).
Reference: C. Keeler, FL, P. Lisbao, arXiv: 1404.1379
3
Setting
• Consider matter in a general theory withN ≥ 2 SUSY.
• In terms ofN = 2 fields: one SUGRA multiplet,N − 2 (massive)gravitini, nV vector multiplets, nH hyper multiplets.
• Setting: focus on extremal black holes where it is sufficient toconsider the AdS2 × S2 near horizon region.
• The final result:
δS =1
12[23− 11(N − 2)− nV + nH ] logAH .
• Example (relevant for microscopics): no correction inN = 4theory with an arbitrary number ofN = 4 matter multiplets.
4
Prelude: Chiral Primaries• Massless fields in AdS2 × S2 organize themselves in short
representations of the SU(2|1, 1) supergroup.
• CFT language: consider chiral multiplets where (h, j) are
(k, k), 2(k +1
2, k − 1
2), (k + 1, k − 1) .
Possible values of k = 12, 1,
32, . . .. (k = 1
2 extra short).
• In the early days of AdS/CFT three groups independently solvedlinearized equations of motion and computed spectra.
• They all found the same spectrum for theN = 2 SUGRAmultiplet.
• We used an indirect argument and found a different result.
• We are right .
5
Spherical Harmonics• Expansion on S2 of single field component with helicity λ: angular
momenta j = |λ|, |λ| + 1, . . ..
• Example: for a gauge field all components organize themselvesinto two towers with j = 1, 2, . . . and two towers with j = 0, 1, . . .
• The physical components of the vector field componentsorganize themselves into two towers with j = 1, 2, . . ..
• So: the set of physical angular momenta in eachN = 2 isunambiguous.
• Example: theN = 2 vector multiplet has one vector field and tworeal scalars so the physical boson towers are: two withj = 1, 2, . . . and two with j = 0, 1, . . ..
• Mixing is allowed (for same j) but assembly of towers into chiralmultiplets uniquely determine conformal weights.
6
The Spectrum of Chiral Primaries
• Result: the spectrum of (h, j) for all chiral primaries:
Supergravity : 2[(k + 2, k + 2) , 2(k + 52, k + 3
2) , (k + 3, k + 1)]Gravitino : 2[(k + 3
2, k + 32) , 2(k + 2, k + 1) , (k + 5
2, k + 12)]
Vector : 2[(k + 1, k + 1) , 2(k + 32, k + 1
2) , (k + 2, k)]Hyper : 2[(k + 1
2, k + 12) , 2(k + 1, k) , (k + 3
2, k −12)]
Each tower has k = 0, 1, . . ..
• Discrepancy: previous work had one more entry in the SUGRAmultiplet
(1, 1) , 2(3
2,1
2), (2, 0) .
• Clarification: this field exists only as a boundary mode.
7
Example: Constraints for Gravity• The graviton in D dimensions has D(D + 1)/2 components, D
gauge symmetries (from diffeomorphisms), D constraints (eom’sleft after gauge fixing).
• So: a graviton has D(D − 3)/2 physical components.
• In 2D a graviton has −1 degrees of freedom so a graviton and ascalar combined has no degrees of freedom.
• Details: after gauge fixing some “equations of motion” are in factconstraints (there are no time derivatives).
• Exception: the constraint is solved by one specific spatial profile(the zero-mode on AdS2) so one boundary degree of freedom canbe freely specified.
• These boundary modes are physical (standard in AdS/CFT).
8
Quantum Fluctuations: Strategy• All contributions from quadratic fluctuations around the classical
geometry take the form
e−W =
∫Dφ e−φΛφ =
1√detΛ
.
• The quantum corrections are encoded in the heat kernel
D(s) = Tr e−sΛ =∑i
e−sλi .
• The effective action becomes
W = −1
2
∫ ∞ε2
ds
sD(s) = −1
2
∫ ∞ε2
ds
s
∫dDxK(s) .
• The constant D0 (or K0) we need is (essentially) the 2ndSeeley-deWitt coefficient or equivalently the trace anomaly of theEM-tensor.
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Simple Heat Kernels in 2D• The heat kernel for a scalar field on S2 is elementary:
KsS(s) =
1
4πa2
∞∑k=0
e−sk(k+1)(2k+1) =1
4πa2s
(1 +
1
3s +
1
15s2 + . . .
).
• A massless scalar field on AdS2 involves a continuous spectrum:
KsA(s) =
1
2πa2
∫ ∞0
e−(p2+14)sp tanhπp dp .
• The local terms in the AdS2 heat kernel is identical to S2 exceptfor the sign of the curvature:
KsA(s) =
1
4πa2s
(1− 1
3s +
1
15s2 + . . .
).
• The heat kernel for a fermion on S2 is also elementary:
KfS(s) =
1
4πa2
∞∑k=0
e−s(k+1)2(2k+2) =1
4πa2s
(1− 1
6s− 1
60s2 + . . .
).
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Simple Heat Kernels on AdS2 × S2
• For a product space heat kernels multiply so for a scalar onAdS2 × S2:
Ks4(s) = Ks
S(s)KsA(s) =
1
16π2a4s2
(1 +
1
45s2 + . . .
).
• For a Dirac fermion on AdS2 × S2:
Kf4 (s) = 4Kf
S(s)KfA(s) = − 1
4π2a4s2
(1− 11
180s2 + . . .
).
• A benchmark for results inN = 2 theory: a “free hyper”
Kmin4 (s) = 4Ks
4(s) + Kf4 (s) =
1
4π2a4s2· 1
12s2 .
• The leading 1/s2 singularity cancels: no cosmological constantfor equal number of fermion and bosons.
• The 1/s order also cancels: this is an accident.
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The AdS2 Perspective• The canonical heat kernel on AdS2 of for a massless field.
• A field with conformal weight h (mass m2 = h(h− 1)) and SU(2)quantum number j (degeneracy 2j + 1):
KA(h, j; s) = KA(h = 1, j = 0; s) e−h(h−1)s(2j + 1) .
• A free 4D boson is a tower of 2D bosons with (h, j) = (k + 1, k)with k = 0, 1, . . . so
Ks4(s) = Ks
A(s) · 1
4πa2
∞∑k=0
e−sk(k+1)(2k + 1)
=1
16π2a4s2
(1 +
1
45s2 + . . .
).
• The sum over the tower of AdS2 fields computes the factor fromthe heat kernel on S2.
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The Vector-Multiplet: Bulk• The conformal weights for fields in supergravity are “shifted” from
the free values.
• The fermions in the vector multiplet are canonical but bosonsinteract: this is the attractor mechanism.
• The “shifted” sum on S2 for all four physical bosons:
KV,b4 (s) =
2KsA(s)
4πa2
∞∑k=0
(e−sk(k+1)(2k + 3) + e−s(k+1)(k+2)(2k + 1)
)=
1
4π2a4s2
(1 +
1
45s2 + . . . +
1
2s(1− 1
3s) + . . .
).
• Heat kernel for the full vector multiplet including fermions:
KV4 (s) =
1
4π2a4
(1
2s− 1
12+ . . .
).
• A 1/s term was generated by interactions.
• The constant term changed sign due to interactions.
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The Hyper-Multiplet• The bosons in the hyper multiplet are canonical – just four free
fields.
• The fermions interact with the graviphoton so the conformalweights differ from a free field.
• The S2 tower of fermions is shifted relative to a free fermion.
• Heat kernel for the complete hyper-multiplet:
KH4 (s) =
1
4π2a4
(−1
s− 1
12+ . . .
).
• A 1/s term was generated by interactions.
• The constant term changed sign due to interactions.
14
The Vector-Multiplet: Boundary• The vector multiplet has a feature not yet discussed: gauge
invariance.
• Two auxiliary towers cancel: unphysical states (violate gaugecondition) and physical (but pure gauge).
• The boundary state: one of the would-be gauge functions is notnormalizable so one state survives.
• Alternatively: one equation of motion is a constraint so onespatial profile survives.
• The boundary state is a massless boson on S2:
−∇IδAI = −∇2Λ = 0
• Final result for the heat kernel:
KV4 (s) =
1
4π2a4
(1
2s− 1
12
)+
1
4π2a4
(1
2s+
1
6
)=
1
4π2a4
(1
s+
1
12
).
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The (Massive) Gravitino Multiplet• Bulk modes: bosons and fermions all have conformal weight
shifted from the free value.
• Boundary modes: two vectors each have a gauge symmetry andso a boundary scalar.
• The SUSY variation is a fermionic gauge symmetry of thegravitino that gives a boundary fermion
γI∇Iε = 0 .
• The boundary heat kernel is constant because of boson-fermiondegeneracy
K(3/2)bndy =
1
4π2a4· 1
2.
• The full heat kernel:
K(3/2) =1
4π2a4·(
(−1
s+
5
12) +
1
2
)=
1
4π2a4·(−1
s+
11
12
)
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The Graviton Multiplet
• Five bosonic boundary modes: four from diffeomorphisms andone from gauge symmetry.
• Boundary modes for diffeomorphisms acquire a mass
(gIJ∇2 + RIJ)ξJ = 0 .
• The S2 vectors have helicity λ = ±1 so angular momentaj = 1, 2, . . .
• The mass of modes due to S2 diffeomorphisms
m2 = k(k + 1)− 2 ; k = 1, 2, . . .
• The mass of modes due to AdS2 diffeomorphisms
m2 = k(k + 1) + 2 ,m2 = k(k + 1) ; k = 0, 1, 2, . . .
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• Four fermionic boundary modes (two preserved SUSYs) withcontribution to mass from background graviphoton
m2 = (k + 1)2 − 1 , k = 0, 1, . . .
• The heat kernel for all boundary modes in the graviton multiplet
Kgravbndy =
1
4π2a4· 52
(1
s+
1
3)− 1
4π2a4
(2
s+
5
3
)=
1
4π2a4
(1
2s− 5
6
)• Bulk modes: bosons and fermions all have conformal weight
shifted from the free value.
• Full heat kernel
Kgrav =1
4π2a4
((
1
2s− 1
12) + (
1
2s− 5
6)
)=
1
4π2a4
(1
s− 11
12
)
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The Quadratic Divergence• Heat kernel for all multiplets, including physical states in bulk and
on boundary
Kphys =1
4π2a4
[(1
s− 11
12) + (N − 2) · (−1
s+
11
12) + nV (
1
s+
1
12) + nH(−1
s− 1
12)
].
• Contributions to the quadratic divergence (the 1/s term):interactions in bulk and counting boundary degrees of freedom.
• Net result: alternating sign.
• Special caseN ≥ 4 theory (with any matter): quadraticdivergence cancels (a consistency check).
• ForN = 3: all divergences cancel for any nV = nH .
• ForN = 2: a new result .
19
4D Zero Modes: General• 4D zero modes: AdS2 boundary states and also massless onS2.
• Physical origin: the global part of each unbroken gaugesymmetry.
• Zero-modes play a special role in the 4D heat kernel:
D(s) =∑i
e−sλi =∑λi 6=0
e−sλi + N0
• The path integral reduces to an ordinary integral
e−W =
∫Dφ0 = Vol[φ0] ∼ ε−N0∆ .
• The correct zero-mode contribution: larger than the naıve resultby a factor of the scaling dimension ∆.
20
4D Zero Modes: Computation
• Vector fields: no new issue since ∆ = 1 for a vector field.
• Bosonic 0-modes in SUGRA multiplet: 6 diff’s on S2 (two withj = 1) and scaling dimension ∆2 = 2. (Heat kernel counts as if∆2 = 1).
• Fermionic 0-modes in SUGRA multiplet: 8 preserved SUSYs∆3/2 = 3
2. (Heat kernel counts as if ∆3/2 = 12).
• Correction due to 0-modes
Kzm =1
8π2a4·[
6 · (2− 1)− 8 · (3
2− 1
2)
]=
1
4π2a4(−1) .
• Note: much of the literature accounts incorrectly for 0-modes.
21
Example: Reissner-Nordstrom
Consider a purely bosonic solution: gravity+Maxwell.
Contributions are the bosonic terms from theN = 2 SUGRAmultiplet:
• Four free bulk bosons (2 gravity + 2 gauge field):δS = − 1
45 logAH .
• Interactions (bulk bosons not quite free): δS = −32 logAH .
• 5 Boundary modes (4 gravity+1 gauge field): δS = −56 logAH .
• Zero-modes: δS = −3 logAH .
Total: δS = −24145 logAH .
(Fermions in SUGRA multiplet add δS = 1309180 logAH)
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Summary
We re-computed quadratic fluctuation determinants around anAdS2 × S2 near horizon geometry.
Some features of our strategy:
• Setting: a general theory withN ≥ 2 SUSY.
• Focus on states that are on-shell.
• Interactions due to background: encoded in chiral primaries.
• Compute also the renormalization of the gravitational couplingconstant (quadratic divergence, 1/s term in the heat kernel).
• Contributions from bulk (4D), Boundary (2D), and Zero-mode(0D).
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